Finite Group Databasebeta

← #6494518

Subgroups — #6494518

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#33287C211yesmaximal, characteristic, complemented
4#33230C211yesmaximal, characteristic, complemented
5#33170C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic, complemented
7#30305C211yesmaximal, characteristic, complemented
8#30261C211yesmaximal, characteristic, complemented
9#29843C211yesmaximal, characteristic
10#29808C211yesmaximal, characteristic, complemented
11#28929C214yesmaximal, complemented
12#28886C214yesmaximal, complemented
13#26980C214yesmaximal, complemented
14#26846C214yesmaximal, complemented
15#22268C211yesmaximal, characteristic, complemented
16#22218C211yesmaximal, characteristic, complemented
17#21691C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#20473C211yesmaximal, characteristic, complemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#489C2214yes—
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#465C2214yes—
26C2×#441C2214yes—
27C2×#431C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2647—21no—
38#2644—21no—
39#2628—21no—
40#2624—21no—
41#2513—21no—
42#2511—21no—
43#2511C2212yes—
44#2492—21no—
45#2492C2212yes—
46#2490—21no—
47#2454—21no—
48#2451—21no—
49#2451C2212yes—
50#2448C2212yes—
51#2447C2212yes—
52#2427C2212yes—
53#2426C2212yes—
54#2423—21no—
55#2423C2212yes—
56#2420—21no—
57#2331—21no—
58#2324—21no—
59#2319—21no—
60#2314—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2084C2214yes—
67#2065C2214yes—
68#2058C2214yes—
69#2056C2214yes—
70#2034C2214yes—
71#2015C2214yes—
72#2015C2214yes—
73#2001C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1971C2214yes—
77#1968C2214yes—
78#1959C2214yes—
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1936C2214yes—
86#1935C2214yes—
87#1935C2214yes—
88#1933C2214yes—
89#1927C2214yes—
90#1892C2214yes—
91#1880C2214yes—
92#1877C2214yes—
93#1865C2214yes—
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1770C2212yes—
99#1770C2212yes—
100#1762C2214yes—
101#1752C2212yes—
102#1752C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C23×C8C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#124C2312yes—
119C2×#124C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×#117D411yescharacteristic
133C2×#117D411yescharacteristic
134C22×C42C2311yescharacteristic
135#494—24no—
136#492—24no—
137#489—48no—
138#487—24no—
139#486—24no—
140#465—48no—
141#441—48no—
142#431—24no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8—24no—
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#400—22no—
166#400—22no—
167#400—22no—
168#399—22no—
169#399—22no—
170#398—22no—
171#398—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#395—22no—
178#394—22no—
179#394—22no—
180#393—22no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—22no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#390—22no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390C2312yes—
203#390C2312yes—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#388—22no—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#386—22no—
230#385—22no—
231#384—21no—
232#384—21no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2312yes—
238#382—21no—
239#382—21no—
240#382—21no—
241#381—22no—
242#380—21no—
243#380—21no—
244#380—21no—
245#380—21no—
246#380—21no—
247#380—22no—
248#380C2312yes—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263#376D411yescharacteristic
264#376D411yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×Q16—28no—
288C2×Q16—44no—
289C2×Q16—44no—
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C22×C8—24no—
294C22×C8—24no—
295C22×C8—42no—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—82no—
309#127—22no—
310#127—22no—
311#125—82no—
312#124—44no—
313#124—44no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—82no—
317#121—44no—
318#121—44no—
319#121—82no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—82no—
323C4×D4—44no—
324C4×D4—44no—
325C4×D4—82no—
326#117—24no—
327#117—42no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#106—44no—
400#106—44no—
401C4⋊C8—24no—
402#103—28no—
403#103—28no—
404#103—28no—
405#103—44no—
406#103—44no—
407#102—28no—
408#102—28no—
409#102—44no—
410C22⋊C8—24no—
411#97—24no—
412#97—24no—
413#97—42no—
414#97—42no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8#4114yes—
419C4×C8#4114yes—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446C24—22no—
447C24C22×D411yescharacteristic
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—82no—
461C2×Q8—82no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—82no—
474C2×D4—82no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—44no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4#13714yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2511yescharacteristic
509Q16—416no—
510SD16—88no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—48no—
521C2×C8C2×#4114yes—
522C2×C8C2×#4114yes—
523C2×C8H(5,𝔽2)14yes—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—21no—
586C42—21no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—82no—
592C42—82no—
593C42—82no—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23C2×#13712yes—
610C23C23×D411yescharacteristic
611Q8—44no—
612Q8—44no—
613Q8—44no—
614Q8—44no—
615Q8—84no—
616Q8—164no—
617D4—44no—
618D4—44no—
619D4—84no—
620D4—84no—
621D4—164no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—28no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—164no—
680C2×C4#53212yes—
681C2×C4#53212yes—
682C2×C4C2×#13712yes—
683C2×C4C2×#13712yes—
684C2×C4#57512yes—
685C2×C4#57712yes—
686C2×C4#57712yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)12yes—
690C2×C4#58512yes—
691C8—24no—
692C8—24no—
693C8—24no—
694C8—42no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22C2×#53211yescharacteristic
710C22#349811yescharacteristic
711C22#350011yescharacteristic
712C22C22×#13711yescharacteristic
713C22C2×#57511yescharacteristic
714C22C2×#57711yescharacteristic
715C22C2×#57711yescharacteristic
716C4—22no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4#317314yes—
735C4#318114yes—
736C4#346012yes—
737C4#349412yes—
738C2—42no—
739C2—162no—
740C2#3527611yescharacteristic
741C2C2×#317311yescharacteristic
742C2C2×#318111yescharacteristic
743C2#6242211yescharacteristic
744C2#6242711yescharacteristic
745C2#6254211yescharacteristic
746C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #35283C2semidirect
3. #33287C2semidirect
4. #33230C2semidirect
6. #33043C2semidirect
7. #30305C2semidirect
8. #30261C2semidirect
10. #29808C2semidirect
11. #28929C2semidirect
12. #28886C2semidirect
13. #26980C2semidirect
14. #26846C2semidirect
15. #22268C2semidirect
16. #22218C2semidirect
17. #21691C2semidirect
18. #21646C2semidirect
19. #20473C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352882
#352832
#332872
#332302
#331702
#330432
#303052
#302612
#298432
#298082
#289292
#288862
#269802
#268462
#222682
#222182
#216912
#216462
#204732